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Questions and Answers
What is the definition of Lesson 6?
What is the definition of Lesson 6?
Solve for x: $4^{3x} = 4^{x-1}$.
Solve for x: $4^{3x} = 4^{x-1}$.
-1/2
Solve for x: $3^{5x+7} = 3^{7x-1}$.
Solve for x: $3^{5x+7} = 3^{7x-1}$.
4
Solve $9^{4x} = 243$ for x.
Solve $9^{4x} = 243$ for x.
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Solve for x: $64 = 2^{2x-4}$.
Solve for x: $64 = 2^{2x-4}$.
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Solve for x: $5^{2x-5} = 125$.
Solve for x: $5^{2x-5} = 125$.
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Solve for x: $4 imes 2^x = 32$.
Solve for x: $4 imes 2^x = 32$.
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Solve $100^{4x} = 1000$ for x.
Solve $100^{4x} = 1000$ for x.
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Solve for x: $e^{2x} = 4$.
Solve for x: $e^{2x} = 4$.
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Solve for x: $5^{-2x-9} = 5^{4x+15}$.
Solve for x: $5^{-2x-9} = 5^{4x+15}$.
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Solve for x: $9^{2x} = 27$.
Solve for x: $9^{2x} = 27$.
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Solve for x: $8^{7x-5} = 8^{4x-3}$.
Solve for x: $8^{7x-5} = 8^{4x-3}$.
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Solve for x: $10^{3x} = 12$.
Solve for x: $10^{3x} = 12$.
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Solve for x: $128 = 2^{3x-5}$.
Solve for x: $128 = 2^{3x-5}$.
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If $7^{6x-11} = 7^{2x+1}$, what is the value of x?
If $7^{6x-11} = 7^{2x+1}$, what is the value of x?
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Solve for x: $8 imes e^{x-1} = 56$.
Solve for x: $8 imes e^{x-1} = 56$.
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Solve for x: $2^{4x} = 6$.
Solve for x: $2^{4x} = 6$.
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Solve for x: $16^{3x} = 64$.
Solve for x: $16^{3x} = 64$.
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What is the definition of Lesson 7?
What is the definition of Lesson 7?
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Which expression is equivalent to $log_4 x^5$?
Which expression is equivalent to $log_4 x^5$?
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Match each logarithm with its equivalent expression.
Match each logarithm with its equivalent expression.
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Which expressions are equivalent to $log_9 14$?
Which expressions are equivalent to $log_9 14$?
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What is the value of $log_{27} 165$?
What is the value of $log_{27} 165$?
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What is the value of $log_6 49$?
What is the value of $log_6 49$?
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Which expression is equivalent to $log_2 6^x$?
Which expression is equivalent to $log_2 6^x$?
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What is the value of $log_{17} 2$?
What is the value of $log_{17} 2$?
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What is the value of $log_5 75$?
What is the value of $log_5 75$?
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Which expression is equivalent to $log_3 24^2$?
Which expression is equivalent to $log_3 24^2$?
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What is the definition of Lesson 8?
What is the definition of Lesson 8?
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If $4^{3x} = 37.9$, what is the value of x?
If $4^{3x} = 37.9$, what is the value of x?
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Solve for x: $5^{x/2} = 65$, round your answer to three places.
Solve for x: $5^{x/2} = 65$, round your answer to three places.
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Solve $6^{3x+1} = 10.8$ for x. Which answer shows the correct steps?
Solve $6^{3x+1} = 10.8$ for x. Which answer shows the correct steps?
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Solve the equation for x: $8^{x+2} = 99.6$.
Solve the equation for x: $8^{x+2} = 99.6$.
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If $3^{(x/2)} + 4 = 7.32$, what is the value of x?
If $3^{(x/2)} + 4 = 7.32$, what is the value of x?
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If $17^{2x-3} = 35$, what is the value of x? Round your answer to three places.
If $17^{2x-3} = 35$, what is the value of x? Round your answer to three places.
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Solve for x: $2^{(x/3)} = 100.6$.
Solve for x: $2^{(x/3)} = 100.6$.
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If $3^{(x/4)} + 2 = 56.8$, what is the value of x?
If $3^{(x/4)} + 2 = 56.8$, what is the value of x?
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Solve $25^{2x-1} = 400$ for x. Which answer shows the correct steps?
Solve $25^{2x-1} = 400$ for x. Which answer shows the correct steps?
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If $6^{4x} = 82$, what is the value of x?
If $6^{4x} = 82$, what is the value of x?
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What is the definition of Lesson 9?
What is the definition of Lesson 9?
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On the first day of a social media ad campaign, a website had 15 unique visitors. Over the following several days, the website received three times the number of unique visitors as the previous day. Which equation can be used to determine the number of unique visitors, v, that the website would have on day d of the ad campaign?
On the first day of a social media ad campaign, a website had 15 unique visitors. Over the following several days, the website received three times the number of unique visitors as the previous day. Which equation can be used to determine the number of unique visitors, v, that the website would have on day d of the ad campaign?
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The membership fee at a sports club was $600 per year. Each year, the membership fee increased by 7%. Which equation models the cost of the membership fee, y, after x years?
The membership fee at a sports club was $600 per year. Each year, the membership fee increased by 7%. Which equation models the cost of the membership fee, y, after x years?
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A nearby pond has 5 frogs, and the population doubles every year. Which inequality, where t is the number of years, models when the population of frogs will be greater than 75?
A nearby pond has 5 frogs, and the population doubles every year. Which inequality, where t is the number of years, models when the population of frogs will be greater than 75?
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On which day of the ad campaign did the website have 10,000 visitors based on the model $v=15⋅3^{(d-1)}$?
On which day of the ad campaign did the website have 10,000 visitors based on the model $v=15⋅3^{(d-1)}$?
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Study Notes
Exponential Equations
- An exponential equation can take the form ( a^x = b ), where ( a ) and ( b ) are constants.
- Example: For ( 4^{3x} = 4^{x - 1} ), solving yields ( x = -\frac{1}{2} ).
Solving for x in Various Equations
- ( 3^{5x + 7} = 3^{7x - 1} ) results in ( x = 4 ).
- For ( 9^{4x} = 243 ), the solution is ( x = \frac{5}{8} ).
- The equation ( 64 = 2^{2x - 4} ) gives ( x = 5 ).
- In ( 5^{2x - 5} = 125 ), ( x ) equals 4.
- The equation ( 4 \cdot 2^x = 32 ) solves to ( x = 3 ).
- ( 100^{4x} = 1000 ) yields ( x = \frac{3}{8} ).
- The equation ( e^{2x} = 4 ) resolves to ( x = \frac{\ln 4}{2} ).
- For ( 5^{-2x - 9} = 5^{4x + 15} ), the solution is ( x = -4 ).
- Solving ( 9^{2x} = 27 ) gives ( x = \frac{3}{4} ).
- Equation ( 8^{7x - 5} = 8^{4x - 3} ) results in ( x = \frac{2}{3} ).
- ( 10^{3x} = 12 ) resolves to ( x = \frac{\log 12}{3} ).
- For ( 128 = 2^{3x - 5} ), the answer is ( x = 4 ).
- If ( 7^{6x - 11} = 7^{2x + 1} ), then ( x = 3 ).
- Solving ( 8 \cdot e^{x - 1} = 56 ) yields ( x = \ln 7 + 1 ).
- The equation ( 2^{4x} = 6 ) simplifies to ( x = \frac{\log_2 6}{4} ).
- For ( 16^{3x} = 64 ), the solution is ( x = \frac{1}{2} ).
Properties of Logarithms
- Logarithmic expressions can be manipulated using properties:
- ( \log_a (x^b) = b \cdot \log_a (x) ).
- Example: ( \log_4 (x^5) ) is equivalent to ( 5 \log_4 (x) ).
- ( \log_7 (y^{10}) ) equals ( 10 \log_7 (y) ).
- Natural logarithms: ( \ln (7^{10}) = 10 \ln (7) ).
- ( \log (7^4) ) can be expressed as ( 4 \log (7) ).
Evaluating Logarithms
- ( \log_9 (14) ) can be calculated as ( \frac{\ln 14}{\ln 9} ) or ( \frac{\log 14}{\log 9} ).
- Calculated values include ( \log_{27} (165) \approx 1.549 ) and ( \log_6 (49) \approx 2.17 ).
- Expression ( \log_2 (6^x) = x \log_2 (6) ).
- Other calculated values: ( \log_{17} (2) \approx 0.24 ) and ( \log_5 (75) \approx 2.68 ).
- ( \log_3 (24^2) ) simplifies to ( 2 \log_3 (24) ).
Solving Exponential Equations with Technology
- Utilizing technology can aid in solving complex exponential equations.
- Example: For ( 4^{3x} = 37.9 ), ( x \approx 0.874 ).
- ( 5^{x/2} = 65 ) results in ( x \approx 5.187 ).
- Equation ( 6^{3x + 1} = 10.8 ) yields ( x \approx 0.109 ).
- If ( 8^{x + 2} = 99.6 ), then ( x \approx 0.213 ).
- For ( 3^{(x/2)} + 4 = 7.32 ), the solution is ( x \approx -4.38 ).
- Equation ( 17^{2x - 3} = 35 ) results in ( x \approx 2.127 ).
- ( 2^{(x/3)} = 100.6 ) leads to ( x \approx 19.96 ).
- Solving ( 3^{(x/4)} + 2 = 56.8 ) gives ( x \approx 6.708 ).
- The equation ( 25^{(2x - 1)} = 400 ) resolves to ( x \approx 1.431 ).
- For ( 6^{4x} = 82 ), the solution is ( x \approx 0.615 ).
Exponentials in Context
- Example of a social media ad campaign: Initial visitors are 15, with growth modeled by ( v = 15 \cdot 3^{(d-1)} ).
- Membership fee at a sports club initiates at $600, increasing by 7% each year, modeled by ( y = 600(1.07)^x ).
- A frog population starts with 5 and doubles each year; the model ( 5(2)^t > 75 ) determines time until the population exceeds 75.
- Identifying when the website reaches 10,000 visitors: Day 7 in the campaign.
- Cost equation for sports club membership demonstrates percentage increase over time.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on exponential equations and their solutions with this flashcard quiz from Algebra 2 B, Unit 2. Perfect for mastering exponential and logarithmic functions, this quiz helps reinforce important concepts and problem-solving skills.
